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G = C2×C3⋊D36order 432 = 24·33

Direct product of C2 and C3⋊D36

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — C2×C3⋊D36
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C9×Dic3 — C3⋊D36 — C2×C3⋊D36
 Lower central C3×C9 — C3×C18 — C2×C3⋊D36
 Upper central C1 — C22

Generators and relations for C2×C3⋊D36
G = < a,b,c,d | a2=b3=c36=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1468 in 194 conjugacy classes, 53 normal (25 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C9, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C3×C9, C36, D18, D18, C2×C18, C2×C18, C3×Dic3, S3×C6, C2×C3⋊S3, C62, C2×D12, C2×C3⋊D4, C3×D9, C9⋊S3, C3×C18, C3×C18, D36, C2×C36, C22×D9, C22×D9, C3⋊D12, C6×Dic3, S3×C2×C6, C22×C3⋊S3, C9×Dic3, C6×D9, C6×D9, C2×C9⋊S3, C2×C9⋊S3, C6×C18, C2×D36, C2×C3⋊D12, C3⋊D36, Dic3×C18, C2×C6×D9, C22×C9⋊S3, C2×C3⋊D36
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, D12, C3⋊D4, C22×S3, D18, S32, C2×D12, C2×C3⋊D4, D36, C22×D9, C3⋊D12, C2×S32, S3×D9, C2×D36, C2×C3⋊D12, C3⋊D36, C2×S3×D9, C2×C3⋊D36

Smallest permutation representation of C2×C3⋊D36
On 72 points
Generators in S72
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)
(1 25 13)(2 14 26)(3 27 15)(4 16 28)(5 29 17)(6 18 30)(7 31 19)(8 20 32)(9 33 21)(10 22 34)(11 35 23)(12 24 36)(37 49 61)(38 62 50)(39 51 63)(40 64 52)(41 53 65)(42 66 54)(43 55 67)(44 68 56)(45 57 69)(46 70 58)(47 59 71)(48 72 60)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
(1 9)(2 8)(3 7)(4 6)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(37 63)(38 62)(39 61)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(64 72)(65 71)(66 70)(67 69)

G:=sub<Sym(72)| (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63), (1,25,13)(2,14,26)(3,27,15)(4,16,28)(5,29,17)(6,18,30)(7,31,19)(8,20,32)(9,33,21)(10,22,34)(11,35,23)(12,24,36)(37,49,61)(38,62,50)(39,51,63)(40,64,52)(41,53,65)(42,66,54)(43,55,67)(44,68,56)(45,57,69)(46,70,58)(47,59,71)(48,72,60), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,9)(2,8)(3,7)(4,6)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(64,72)(65,71)(66,70)(67,69)>;

G:=Group( (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63), (1,25,13)(2,14,26)(3,27,15)(4,16,28)(5,29,17)(6,18,30)(7,31,19)(8,20,32)(9,33,21)(10,22,34)(11,35,23)(12,24,36)(37,49,61)(38,62,50)(39,51,63)(40,64,52)(41,53,65)(42,66,54)(43,55,67)(44,68,56)(45,57,69)(46,70,58)(47,59,71)(48,72,60), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,9)(2,8)(3,7)(4,6)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(37,63)(38,62)(39,61)(40,60)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(64,72)(65,71)(66,70)(67,69) );

G=PermutationGroup([[(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,61),(35,62),(36,63)], [(1,25,13),(2,14,26),(3,27,15),(4,16,28),(5,29,17),(6,18,30),(7,31,19),(8,20,32),(9,33,21),(10,22,34),(11,35,23),(12,24,36),(37,49,61),(38,62,50),(39,51,63),(40,64,52),(41,53,65),(42,66,54),(43,55,67),(44,68,56),(45,57,69),(46,70,58),(47,59,71),(48,72,60)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,9),(2,8),(3,7),(4,6),(10,36),(11,35),(12,34),(13,33),(14,32),(15,31),(16,30),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24),(37,63),(38,62),(39,61),(40,60),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(64,72),(65,71),(66,70),(67,69)]])

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 18A ··· 18I 18J ··· 18R 36A ··· 36L order 1 2 2 2 2 2 2 2 3 3 3 4 4 6 ··· 6 6 6 6 6 6 6 6 9 9 9 9 9 9 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 18 18 54 54 2 2 4 6 6 2 ··· 2 4 4 4 18 18 18 18 2 2 2 4 4 4 6 6 6 6 2 ··· 2 4 ··· 4 6 ··· 6

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 D6 D9 C3⋊D4 D12 D18 D18 D36 S32 C3⋊D12 C2×S32 S3×D9 C3⋊D36 C2×S3×D9 kernel C2×C3⋊D36 C3⋊D36 Dic3×C18 C2×C6×D9 C22×C9⋊S3 C22×D9 C6×Dic3 C3×C18 D18 C2×C18 C3×Dic3 C62 C2×Dic3 C18 C3×C6 Dic3 C2×C6 C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 4 1 1 1 1 1 2 2 1 2 1 3 4 4 6 3 12 1 2 1 3 6 3

Matrix representation of C2×C3⋊D36 in GL6(𝔽37)

 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 36 36 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 22 34 0 0 0 0 26 15 0 0 0 0 0 0 36 0 0 0 0 0 1 1 0 0 0 0 0 0 6 26 0 0 0 0 11 17
,
 1 0 0 0 0 0 27 36 0 0 0 0 0 0 1 0 0 0 0 0 36 36 0 0 0 0 0 0 31 11 0 0 0 0 17 6

G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,1,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,26,0,0,0,0,34,15,0,0,0,0,0,0,36,1,0,0,0,0,0,1,0,0,0,0,0,0,6,11,0,0,0,0,26,17],[1,27,0,0,0,0,0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,36,0,0,0,0,0,0,31,17,0,0,0,0,11,6] >;

C2×C3⋊D36 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes D_{36}
% in TeX

G:=Group("C2xC3:D36");
// GroupNames label

G:=SmallGroup(432,307);
// by ID

G=gap.SmallGroup(432,307);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^36=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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